PAMM · Proc. Appl. Math. Mech. 10, 373 – 374 (2010) / DOI 10.1002/pamm.201010179

A Multiphasic Model Describing the CO2 Injection Problem

Irina Komarova1,∗ and Wolfgang Ehlers1

1 Institute of Applied Mechanics (CE), Pfaffenwaldring 7, 70569 Stuttgart / Germany

Internet: http://www.mechbau.uni-stuttgart.de/ls2

According to the annual data, the portion of ejected carbon dioxide (CO2) into the atmosphere is much higher in comparison

to other greenhouse gases. Therefore, the development of various scenarios in order to reduce the CO2 concentration in the

atmosphere is nowadays a challenge and a relevant subject in research. The current investigation is dedicated to the modelling

of a CO2 injection into a water-saturated aquifer as the most capable reservoir, where its upward migration is blocked by

a dense cap-rock layer. The major part of the study is focused on two specific processes taking place in the stored reser-

voir: the phase transfer phenomenon of the supercritical CO2 which occurs under changeable surrounding conditions and

the sequestration-driven deformation of the solid skeleton acting under the pressure of the pumped CO2. For the numerical

simulations within a continuum-mechanical framework, a multiphasic model based on the Theory of Porous Media is intro-

duced. Moreover, the main principles of developing a constitutive equation for the mass production term for the mass balance

equation are discussed here.

c© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

There are different ideas and scenarios developed in the present, which can help to reduce CO2 emissions into the atmosphere

[1]. Besides other possibilities, one of the ways to reduce the amount of the CO2 in the atmosphere is its disposal into saline

aquifers. It is obvious that injection of CO2 will rise the variety of coupled physical, chemical and mechanical processes in the

reservoir. The current investigation is focused only on some of them, such as the solid deformation and the phase transition

phenomenon. These processes could be well described using the continuum-mechanical framework of the Theory of Porous

Media [2]. Herein, the theoretical description of the model is introduced with a particular attention on the specific constitutive

equations for the considered CO2 injection problem.

2 Theoretical Framework

Following the basic concepts of the Theory of Porous Media (TPM), the properties of a multiphasic aggregate with several

individual constituents ϕα with α ∈ {S: solid, β: fluids} are based on a statistical distribution of the components over a

representative elementary volume. Within the TPM, the deformation of the solid matrix ϕS is given within a Lagrangean

description, whereas the fluid constituents ϕβ are described by a modified Eulerian description, where the seepage velocities

of the fluid phases are represented with respect to the velocity of the solid constituent.

supercriticalregion

triple

criticalpoint

point

gaseous

liquid

solid

θ [K]θtr = 216.6 θc = 304.18

0.5

7.38

p [MPa]

Fig. 1 CO2 phase diagram.

The examined porous media model consists of a solid skeleton percolated

by several fluid phases. More precisely, two different approaches regard-

ing the number of the fluid phases are proposed for a consideration. Both

models consist of a solid constituent ϕS represented by the rock and a liq-

uid phase ϕW represented by the water in the aquifer. The difference be-

tween the two formulations is given by the description of the CO2 state.

In the frame of the first formulation, the CO2 occurs only in the supercrit-

ical phase ϕL without any phase transition, while the second formulation

includes phase transition by the possibility for the CO2 to be divided into

two phases: liquid or gaseous ϕG and supercritical ϕL, cf. Fig. 1. The split-

ting of the CO2 into two different phases can occur during the propagation

in the aquifer caused by a change of the surrounding conditions (p, θ). This

leads to the fact that CO2 can change its phase.

Therefore, the question arises how to describe the phase-exchange process carefully. This problem can be solved by providing

an additional constitutive equation for the mass production term, which defines the amount of the supercritical CO2 transferred

into the gaseous CO2 over time.

Within the modelling of coupled multiphasic problems, the behaviour of each constituent can be mathematically described

by primary variables. Usually, as a kinematic primary variable for the solid skeleton ϕS , the displacement vector uS is chosen.

For the fluid phase ϕβ , the pressure usually is taken as primary variable, whereas, for the temperature dependent problem, the

temperature θ is taken into account. The general form of the corresponding governing equations (momentum balance, mass

∗ Corresponding author e-mail: komarova@mechbau.uni-stuttgart.de, Phone: +49 711 685 - 69253, Fax: +49 711 685 - 66347

c© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

374 Section 7: Coupled Problems

balance and energy balance) can be found, for example, in [2]. Further on, by evaluation of the entropy inequality, most of

the constitutive relations which are necessary to close the system of the equation could be derived, cf. [4]. Other constitutive

relations could be developed separately using, for example, the basic thermodynamical principles. Within this contribution,

only some specific constitutive constraints are introduced.

More precisely, the materially incompressible solid exhibits elastic properties. The momentum and energy production

terms differ for the two approaches caused by the mass production term ρG. The seepage velocities for the fluid phases are

described by modified Darcy law. The liquid phase represented by water is materially incompressible, while the other fluid

phases represented by supercritical or gaseous CO2 are materially compressible and described by the Peng-Robinson equation

of state [3]:

pGR =RG θ

ν − B−

A αω

ν2 + 2 B ν − B2. (1)

Therein, A and B are constants depending on the critical properties of CO2, ν is the specific volume, RG = RG/MG is the

specific gas constant, and αω is a constant depending on the acentric factor ω. The capillary-pressure-saturation relationship

between the fluid phases is presented by the van Genuchten law, [4] . Additionally, the constitutive equation for the mass

production is developed. In order to derive this constitutive relation, the entropy balance is used:

ρα(ηα)′α = div (−1

θq

α) +1

θρα rα + ζα . (2)

In (2), ηα is the entropy, qα is the heat flux, which can be calculated by Fourier’s law, rα is the external loading (radiation),

which is set to zero here, ζα = ηα − ραηα is the entropy production term and∑

α ηα ≥ 0. For a reversible process, as

provided in this investigation, a strong assumption can be used stating that the sum of the entropy productions terms is equal

to zero:∑

α ηα = 0. Under this restriction, the relation for the mass production term ρG can be easily found as

ρG =1

ηG − ηL

∑

α

(

−div (1

θq

α) − ρα(ηα)′α +1

θραrα

)

. (3)

The numerical treatment of the problem is carried out by the implementation into the Finite-Element solver PANDAS

(Porous media Adaptive Nonlinear finite element solver based on Differential Algebraic Systems).

3 Carbon Dioxide Injection into an Aquifer

The presented numerical example (Fig. 2) shows the influence of the distribution of the CO2 plume within the reservoir on the

deformation of the solid skeleton within the cap-rock layer and aquifer excluding the possible effects from phase transition

process.

Herein, the CO2 is injected from the left side into the aquifer

at the supercritical state with a density of ρ = 479 [kg/m3]. The

aquifer has the permeability KS0S = 10−7 [m2], while the perme-

ability of the cap-rock is KS0S = 10−13 [m2]. The Lame con-

stants of the solid skeleton are µS = 5.58 · 107 [kN/m2], λS =8.36 · 107 [kN/m2] corresponding to sand. As a result of the simula-

tion, the distribution of the liquid saturation is illustrated in the Fig. 2.

More specifically, a propagation of the CO2 within the aquifer caused

by the buoyancy forces is shown on the picture. The CO2 is blocked

by the nearly dense cap-rock layer and stays within the reservoir.

Further investigation will be focused on the implementation of the

phase-exchange process between CO2 phases into the model using

the constitutive equation presented above.

0.1

0.5

0.98

cap-rock

aquifer

CO2

sL

Fig. 2 CO2 injection into aquifer.

References

[1] B. Metz, O. Davidson, H. C. de Coninck, M. Loos and L. A. Meyer (eds.), Carbon Dioxide Capture and Storage (Cambridge University

Press, Cambridge, 2005), p. 442.

[2] W. Ehlers, International Journal of Advances in Engineering Sciences and Applied Mathematics 1, 1–24 (2009).

[3] D.-Y. Peng and D. B. Robinson, Industrial and Engineering Chemistry Fundamentals 15, 59–64 (1976).

[4] T. Graf, Multiphasic flow processes in deformable porous media under consideration of fluid phase transitions. Dissertation, Report

No. II-17 of the Institute of Applied Mechanics (CE), University of Stuttgart 2008.

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